Properties

Label 56784bk
Number of curves $6$
Conductor $56784$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("bk1")
 
E.isogeny_class()
 

Elliptic curves in class 56784bk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
56784.bh6 56784bk1 \([0, -1, 0, 2648, 10672]\) \(103823/63\) \(-1245548408832\) \([2]\) \(73728\) \(1.0101\) \(\Gamma_0(N)\)-optimal
56784.bh5 56784bk2 \([0, -1, 0, -10872, 97200]\) \(7189057/3969\) \(78469549756416\) \([2, 2]\) \(147456\) \(1.3567\)  
56784.bh3 56784bk3 \([0, -1, 0, -105512, -13076688]\) \(6570725617/45927\) \(908004790038528\) \([2]\) \(294912\) \(1.7033\)  
56784.bh2 56784bk4 \([0, -1, 0, -132552, 18592560]\) \(13027640977/21609\) \(427223104229376\) \([2, 2]\) \(294912\) \(1.7033\)  
56784.bh4 56784bk5 \([0, -1, 0, -91992, 30144048]\) \(-4354703137/17294403\) \(-341920891084910592\) \([2]\) \(589824\) \(2.0498\)  
56784.bh1 56784bk6 \([0, -1, 0, -2119992, 1188797232]\) \(53297461115137/147\) \(2906279620608\) \([2]\) \(589824\) \(2.0498\)  

Rank

sage: E.rank()
 

The elliptic curves in class 56784bk have rank \(0\).

Complex multiplication

The elliptic curves in class 56784bk do not have complex multiplication.

Modular form 56784.2.a.bk

sage: E.q_eigenform(10)
 
\(q - q^{3} + 2 q^{5} - q^{7} + q^{9} + 4 q^{11} - 2 q^{15} - 6 q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.